Maths: Should English Schools Look to Switzerland Rather Than Shanghai For Inspiration?

by Mark Boylan, Sheffield Hallam University, The Conversation: https://theconversation.com/maths-should-english-schools-look-to-switzerland-rather-than-shanghai-for-inspiration-110107

Pexels

When it comes to education, how countries rank against each other in league tables has become a big influence on education policy. And one of the biggest rankings is the Programme for International Student Assessment – also known as PISA. This is a regular comparison of the performance of 15-year-olds in different countries, including in maths.

Increasingly politicians want to see their countries rise up the league tables, which has led to attempts to import ways of teaching from overseas. In England, many schools have adopted East Asian methods in maths teaching to try to climb the league tables. This has come be to be called the “mastery maths” model.

Singapore and Shanghai both do very well in international PISA tests compared to England, but the way both places approach mastery is slightly different. The mastery method adopted in English schools has mostly been modelled on the Shanghai system – and has involved teachers from England going to Shanghai and having Shanghai teachers go to England.

We have spent the past four years researching whether the Shanghai exchange would lead to changes in teaching in England and whether this would lead to better test scores. We found that in many of the schools that were involved in the first exchange in 2014-2015 there have been a lot of changes – and many adopted Shanghai-style maths teaching. These changes have involved slowing down the curriculum, using more learning by heart, more interaction between teacher and pupils, and using different ways of representing maths ideas. Schools have also adopted ways to give all children access to challenging maths.

But when we compared the test scores to similar schools who have not adopted the mastery method, we found there had been no change in the test results of 11-year-olds. For seven-year-olds we found a small change but nothing that would suggest a massive improvement in scores from the new mastery method.

Importing education

Perhaps this isn’t surprising though, because teaching methods are not the only reason for East Asian success. And in some ways, Shanghai maths hasn’t yet had a fair test in England. UK teachers have been expected to adopt Shanghai approaches, but have not had the same amount of time for professional development or to plan lessons.

There’s also the fact that in Shanghai primary schools, children are generally taught by the same teacher for two to three years – so the teacher gets to know the pupils very well. Policies in China that have led to many parents only having one child also mean that young children get a lot of adult attention from parents and grandparents – which could also help their development.

---

Read more: Maths challenge: England has one of the biggest gaps between high and low performing pupils in the developed world

---

Although Shanghai maths imported to England has not had the effect on results that was hoped for, it does not mean that with more time it cannot lead to improvements. And there are particular ideas in the Shanghai maths approach – such as high quality mathematical talk and careful choice of how to represent maths ideas – that do have evidence for success in England as well as other countries.

Further afield

If East Asian methods do not import easily because of other factors, then perhaps politicians should broaden where they look for ideas in maths. Switzerland, for example, was the top European performer in maths in PISA in 2015 – not far behind East Asian countries.

Canada also did significantly better than England. Though recently, Canadians have also started worrying about their performance.

Children as young as two are grouped by ability in English nurseries. Shutterstock

There’s also more that can be learnt from PISA and other international tests. It seems, for example, that all high-performing education systems have some common features. These include: supporting teachers with good pay, conditions and status. Teachers that are given the freedom to decide what and how to teach with support from researchers but without political interference, also seems to be another contributing factor. As does having an emphasis on good outcomes for everyone – not just the highest performing children – most high performing countries do not group children by ability from a young age, as often happens in England.

In any case, different tests tell different stories. Another international test – Trends in Maths and Science study – shows England’s results in maths have been steadily improving for the past 20 years. Perhaps then, the most important lesson from England’s mastery experiment is to stop worrying so much about international league table positions – and for schools to focus on encouraging all pupils to be the best they can be.

---

Read more: Why global education rankings don't reveal the whole picture

---

Mark Boylan, Professor of Education, Sheffield Hallam University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

Here's How to Get Kids to Remember Times Tables

chartmedia.co.uk

by Craig Speelman, Edith Cowan University, The Conversation: http://theconversation.com/heres-how-to-get-kids-to-remember-times-tables-40471

Lots of kids have trouble remembering their times tables. Learning them by rote can mean a child can accurately recite the times tables, but has no idea what the numbers actually mean or how to apply this knowledge in a maths problem.

Practice is essential to effective learning, but it is important to keep a balance between practice and application.

Children need to know why they need to learn times tables

The number system underlying the times tables can often seem fairly arbitrary. A child can be forgiven for thinking it’s just a complex system of numbers that they have to learn because the teacher says so.

If you have no idea why you are required to learn something, it is very difficult to develop sufficient motivation to persist with the practice that is often necessary to master the material.

One way to demonstrate the usefulness of the times tables is to engage a child in a counting task. The task could be timed, such that determining the number of elements quickly will mean something for the ultimate result (for example, beating a time limit results in a positive outcome, failing to beat the limit means the task starts again).

The multiplication facts in the times tables can then be demonstrated as short cuts in the counting process (if you can arrange the elements into four groups of eight, then knowing the answer to “4 x 8 = ?” will result in faster performance than having to rely on counting all of the 32 elements).

Many computer games and apps (like IXL Maths) possess this feature. They also involve many other features designed to maintain the interest of a child, which can help keep them motivated enough to persist with the task. Ultimately, the more practice, the better the knowledge.

Get to know the sums individually, not as a song lyric

Memory can often be a good reflection of what we do. If we regularly sing along to a favourite song, each line tends to remind us of the next line. However, if we then try to sing the song by ourselves, without the aid of an accompanying recording, we often find that forgetting one line means subsequent lines also can’t be recalled.

A similar thing can happen if we engage in rote recitation of the times tables. This method is only useful if we want to have a method to fall back upon when all other methods fail. Basically this method can only produce the equivalent of a song lyric where, remembering what “4 8s” are is only possible if you can remember “4 6s are 24” and “4 7s are 28”.

A better form of knowledge is one where a child knows the answer to each multiplication problem as soon as they see it, much like being able to read a word as soon as you see it.

Knowing the answer to each problem is then independent of knowing the answer to other times table problems. This type of knowledge can be gained only by practice at producing the answer.

One method for undertaking this type of practice is something like the old flash-card method. Write a problem on one side of a card (4 x 8 = ?), and the answer on the other side. With a shuffled deck of cards representing all of the problems in the times tables, a child can practise producing the answer to each problem, and then check their response by turning over the card.

Occasionally an adult can ask the child to do this out loud to ensure they are doing the task correctly. Initially the child may have to guess the correct answer, or work it out with their fingers or some other method. But they always have the benefit of immediate feedback by turning over the card.

Eventually, with enough practice, the constant association of the problem with the correct answer will begin to stick in their memory. A similar method can be easily programmed on to a computer or tablet. Plenty of commercial apps are available that will mimic this procedure.

Apply the times tables knowledge

Knowledge of the times tables is not useful by itself. A child must learn to apply the knowledge in a mathematical context.

It is important, though, that a child’s knowledge of the times tables is not allowed to remain as a list of independent facts. A child needs to engage in activities that demonstrate the connections between the multiplication facts in the times tables. It is important to see how 4 x 8 and 8 x 4 are connected.

Ultimately they will also need to see how 4 x 8 = ? and 32 ÷ 8 = ? are connected. To achieve this the child should be provided with activities that require the application of their arithmetic knowledge in a way that can demonstrate and lead the child to uncover these connections.

Practice with this sort of material can help kids develop a knowledge base that results in reliable retrieval of facts and the sort of flexible application of this knowledge that is required in higher-order problems, such as solving for x in 2x + 3 = 11. If you struggle to come up with an answer to this problem, I would not suggest relying on a times tables song to help you out.


This article was originally published on The Conversation. Read the original article.

How a Second Language Trains Your Brain for Math

(Photo: alternative_illustrations/Flickr)

by Nathan Collins, Pacific Standard: http://www.psmag.com/navigation/health-and-behavior/language-trains-brain-math-91289

Speaking more than one language has some advantages beyond ordering food in the local tongue - psychologists believe that bilingualism has many other positive side effects. 

Now, researchers have evidence connecting bilinguals’ talents to stronger so-called executive control in the brain.

Much has been made recently about growing up learning more than one language, as about one in five do.

There’s evidence that children who grow up speaking two languages may be more creative, that bilingualism might stave off dementia, and that bilinguals are better at tasks that involve switching attention between different objects.

That led some researchers to suspect that speaking two languages might improve our brains’ executive functions, the high-level circuits that control our ability to switch between tasks, among other things.

To get a little more sense of the matter, Andrea Stocco and Chantel Prat of the University of Washington screened 17 bilingual and 14 monolingual people for language proficiency and other factors and then tested them using a series of arithmetic problems.

Each problem was defined by a set of operations and two inputs - divide x by two, add one to y, and subtract the second result from the first, for example - with x and y specified uniquely for each problem.

First, participants ran through 40 practice problems using just two operation sets. Next, they went through another 40 problems, this time a mix of 20 new ones, each with a unique set of operations and inputs, and another 20 featuring the previously studied arithmetic operations, but with new inputs for x and y. Finally, the groups worked through 40 more problems, again a mix of familiar and novel, but this time, they completed them inside a fMRI brain scanner.

While bilinguals and monolinguals solved the problems with equal accuracy and took about the same amount of time on arithmetic with familiar sets of operations, bilinguals beat out monolinguals, on average, by about half a second on novel problems.

What’s more, fMRI results showed that the basal ganglia, a brain region previously linked to learning about rewards and motor functions, responded more to novel math problems than old ones, but only in bilinguals.

That’s interesting, Stocco says, because more recent studies suggest the basal ganglia’s real role is to take information and prioritize it before passing it on to the prefrontal cortex, which then processes the information.

If that’s correct, the new results suggest that learning multiple languages trains the basal ganglia to switch more efficiently between the rules and vocabulary of different languages, and these are skills it can then transfer to other domains such as arithmetic.

“Language is one of the hardest things the brain does,” Prat says, though we often realize that only when we try to learn a new language - a task that is “at least an order of magnitude” more difficult than learning the first one.

But just as working on your core has benefits outside the gym, working your basal ganglia hard may be the key to promoting other cognitive skills, especially your hidden math genius.

Nathan Collins studied astrophysics and political science before realizing he wanted to learn about all of the science without worrying about tenure. In his second life as a freelance science writer, he’s written for Scientific American, New Scientist, and others.

Maths Boffins Solve Supermarket BOGOF Quandary

by News Biscuit: http://www.newsbiscuit.com/2014/08/13/maths-boffins-solve-supermarket-bogof-quandary/

Mathematicians at the University of Leicester have proved conclusively that whenever supermarkets use a ‘Buy One, Get One Free’ offer they should place an even number of items on the relevant shelf.

The study, using what the mathematicians called ‘some pretty advanced string and brown paper theory’ has finally resolved the long-standing conundrum that baffled supermarkets as to why they always had one item left at the end of the day.

‘We were always a bit confused about the reasons for this and thought it might be down to customer indifference,’ said Sainsbury’s new CEO Mike Coupe.

‘Now, thanks to the findings of Professor Keith Turner and his team, we can screw our suppliers even further, decimate the few remaining High Streets and inflate our bonuses even further … um, sorry, wrong document … improve our customer service even further.’

Consumer watchdog ‘Which?’ also welcomed the findings, saying that it was something many of its members knew intuitively but it was good to have it confirmed by scientists.

Tesco added that it was also very interested in the result, saying that in the daily battle for a bigger share of the customer market ‘Every Little Helps’. Waitrose, however, tutted slightly and said ‘Really!’, while Netto threatened reporters with a broken bottle before falling over asleep in a car park.

A senior official at the Ministry of the Environment congratulated the Leicester team on their findings, claiming that it could help reduce food waste in Britain by anything up to 50,000 tonnes per day.

Ed Miliband disputed the figure and said that when Labour returns to power after the next election, new legislation will make it illegal to stack shelves with an odd number of items.

The applications of this work are exciting the maths world. Already, a pair of Russian mathematicians is applying the findings to solve the age-old problem of the odd sock left in the drawer.

The team from Leicester will be turning their minds to the solution of another supermarket related problem. Said Professor Turner: ‘We’ll be using the mathematics of the infinitesimally small to calculate, to the nearest nano-penny, the true value of a single Nectar point.’

Coffeemate

Creating a Classroom Culture for Student Success

Math (Photo credit: Wikipedia)

by Gay Clyburn, Carnegie Commons: http://commons.carnegiefoundation.org/what-we-are-learning/2014/creating-a-classroom-culture-for-student-success/

When students walk into developmental math classes they are most likely carrying something weightier than their backpacks, something much more insidious.

They bring with them negative mindsets that they can’t do math or that they aren’t a math person, reinforced by the history of past math classes where they experienced failures.

And many bring with them the threat of stereotypes, some math-based and others defined by gender or ethnicity.

In designing two alternative mathematics pathways for students who place into college developmental math classes, Carnegie has acknowledged this student baggage as one of the key drivers that must be addressed in order to fully support student success.

And they have embedded interventions into the instructional design of the two Pathways - Statway in statistics and Quantway in quantitative reasoning - to address these drivers.

At the annual Community College Pathways National Forum, Claude Steele, the leading expert on stereotype threat, and David Yeager, whose work on transforming student mindsets has been incorporated into the Pathways instructional system since the initial design, suggested approaches that in other settings had been shown to reverse the roles these threats play in negatively affecting the motivation and engagement of students, and thereby their educational outcomes and performance.

As Steele explained, these influences are “powerful but not determinative.”

Steele provided an example of stereotype threat especially relevant to Carnegie’s work. Female and male students who excelled at math at the University of Michigan were administered the half hour section of the graduate math exam.

The premise was that the gender stereotype that females weren’t as good at math as males would suppress the performance of the female students, not allowing them to do the necessary cognitive work on the exam that they were clearly capable of doing. In this case, it held true.

The female students significantly underperformed compared to the male students on the same test. Although both men and women were stressed merely by having to take a test, women experienced the additional pressure of the stereotype.

To mitigate the stereotype - “the preoccupying presence” as Steele puts it - students equally as gifted as the first tested cohort were told beforehand that this particular test was one on which women always did well.

Under these conditions, the female students’ performance increased to match that of men. Similar impacts have been observed for various racial/ethnic groups as well.

Steele suggests remedies for stereotype threat. These include changing the cues that educators send to students. Changing the language in a classroom can create relationships between students, advisors, and teachers that tell a student that there is no presumption that he/she lacks something needed to succeed. Instead, there is a presumption that the student can succeed.

And the idea that ability is malleable is a tremendous relief to kids, and a liberating idea, Steele said. “It significantly reduces stereotype judgment.”

Carnegie is introducing the idea of malleability in the Pathways. One of the exercises in the Pathways Starting Strong package is an exercise where students read an article explaining that neuroscience shows that the brain is like a muscle and that with enough effort they can grow their brain.

Through the Starting Strong package, students who come to the Pathways thinking that they aren’t “math people,” or they don’t belong because they aren’t smart enough to succeed are supported in developing a “growth mindset.”

In addition to learning from the article that intelligence is not fixed, students are given strategies to support persistence through the course, and the encouragement from the start and throughout the course that gives them the courage to use the strategies to succeed.

Yeager said that in a randomized control trial the introduction of this one article on the concept of brain growth has been shown to have a significant effect on student persistence and success. There is evidence from selected Pathways classrooms that indicate that the effect has been replicated through the Starting Strong activities as well.

The Pathways - which subsume rigorous materials, new and more engaging pedagogies, the productive persistence interventions like the use of this exercise, and the behavior and speech that support it - have produced amazing results.

Students have tripled their success rates in half the time and Carnegie has been able to maintain this level of student accomplishment, even as the initiative has grown to include new colleges, new faculty, and many more students over the past three years.

Yeager offered some specific recommendations to those attending this year’s Forum and just beginning to teach the Pathways. He said to create a class culture that supports success, not one that implies expectations of failure.

He said to provide praise after accomplishment (not disassociated from effort and accomplishment), encouragement often, and continuous feedback - in class, during office hours, through emails.

He said to continue to remind students that the brain is analogous to a muscle, that “the more you use it, the better it works.” Or, “the more you practice, the smarter you become.”

When students seem to get discouraged, give them a boost - indeed, there are “booster” activities included in the pedagogy that Pathway faculty use. Use phrases like: “other students say that when you come to the difficult part where you have to struggle, it is a particularly helpful and productive part of the learning process” or “when you struggle, then you’re growing.”

Yeager said that the really wonderful thing about what he had discovered in his research is that the lowest achievers change the most and become some of our highest achievers.

Yeager concluded with a challenge: “Students have theories about their success,” Yeager said. “It is up to us to shift those theories” to more positive and productive ones.

Gay Clyburn is associate vice president, public affairs and is responsible for external relations, marketing and communications, publishing, electronic communications and special events for the Carnegie Foundation.

Better at Reading Than Maths? Don't Blame it All on Your Genes

Mathematics Textbook (Photo credit: Terriko)

by Kathryn Asbury, University of York

I disliked and feared maths for most of my school career and dropped it as soon as I possibly could.

My mother recalls me crying as a five-year-old because: “I can’t do the people-on-the-bus sums”. If the bus has 12 passengers and three get off, how many are left?

English, by contrast, was a breeze.

At seven, I stood on a chair with a microphone and read my version of Sleeping Beauty aloud to the entire school. Reading and writing already ranked high among my passions.

Mine isn’t an unfamiliar tale. Many people label themselves as “not a maths person” or “not much of a reader”, often while they are still children.

And yet, in a recent study published in Nature Communications, scientists showed that around half of the genes that affect how well 12-year-olds in the UK perform in maths also affect how good they are at reading. And they showed this in a new and important way.

For the first time ever, this study - led by UCL’s Oliver Davis, Chris Spencer at Oxford and Robert Plomin at King’s College London - was able to estimate genetic influences on learning abilities using DNA alone.

The implications of this for future genetically sensitive research in the behavioural and social sciences are highly significant. It is certainly much easier to get hold of DNA than it is to get hold of a results from a large twin sample - another good way of researching this area.

Overlapping genes for maths and reading

The researchers analysed millions of DNA variations from almost 3,000 people and found that more than half of the differences between how well 12-year-olds performed in reading and maths could be explained by differences in their genes.

But they also found that reading and maths are correlated partly for genetic reasons: many genes appear to operate in both domains.

These results are not in themselves new. Twin studies have previously reached very similar conclusions. However, the fact that such findings have now been confirmed by a genome-wide association study - where many common genetic variants are explored for association with a particular trait - is a very significant development.

It is interesting that in spite of using DNA data no particular genes emerged as significant influences on reading or maths. Instead, the researchers found collections of subtle DNA variations.

This ties in with other research that has led to an understanding that many genes of small effect combine to influence complex traits and that the effects are mostly too small to pick up reliably, even with large samples. With genes it seems we’re more likely to find teams than star players.

The research supports the Generalist Genes Hypothesis, the idea that genes are generalists and environments are specialists. Twin studies have found a great deal of evidence that important environments are likely to be specific to particular traits or learning outcomes.

For example, a good English teacher might have a slightly different profile to a good maths teacher. The two subjects might benefit from different approaches to homework or different classroom organisation. They may, in short, need to offer different ways of drawing out genetic potential.

Environment still hugely important

What we have here is compelling evidence that genes influence maths and reading abilities and that at least half of the genes influencing one ability also influence the other. So, in that case, how can it be that some kids are so much better at reading than maths, and vice versa?

The first answer is simply that genes do not determine behaviour. Genes offer probabilities rather than prophecies. They represent “what is” rather than “what could be”. Even if you have the capacity to do well in a subject this does not automatically translate to high achievement. Motivation, confidence and interest all have a role to play too.

The second answer is that the genetic overlap between the two skill-sets is not 100%. Although there is evidence for shared genetic effects across reading and maths there is evidence of some genetic specificity too.

Most importantly, this new study highlights the role of the environment. It would appear that our life experiences have a particularly important part to play in making some people better at one subject than the other.

This may happen through a variety of mechanisms including sparking an interest, inspiring future aspirations or nurturing appetites as well as aptitudes. Both genes and experiences influence the choices young people make about further education and careers and cannot be considered in isolation from each other.

The UK government has been pushing recently to improve national mathematics performance, and bring them up to levels seen in East Asia. Shared genetic influence on reading and maths suggests that if average maths performance lags behind average reading performance then this should, from a biological point of view, be possible.

The key to achieving this, however, lies in our environment rather than in our DNA. We need a better understanding of which aspects of the maths learning environment promote traits such as self-efficacy, interest and effort as well as achievement - and which don’t.

We also need to understand that the same approach will not work for everyone. Drawing out individual potential requires at least a degree of personalisation.

And, just for the record, yes I can do the “people-on-the-bus” sums now.


Kathryn Asbury does not work for, consult to, own shares in or receive funding from any company or organisation that would benefit from this article, and has no relevant affiliations.

This article was originally published on The Conversation. Read the original article.

A Lack of Maths Just Doesn't Add Up for a Career in Science

Mathematics Workshop (Photo credit: flgr)

by Trevor Hambley, University of Sydney

Our future in science, technology and engineering relies on a foundation and understanding of mathematics.

And while it is pleasing to see a growth in interest in our advanced mathematics course at the University of Sydney, it is also worrying to see an increase in the number of highly capable students who come from high school with a limited background in mathematics.

This seriously limits their ability to undertake university degrees in science, technology, engineering and mathematical (STEM) areas, even allowing for the availability of bridging courses.

How we support these students entering STEM degree programs with a lower level of mathematical knowledge than is needed is being addressed at a two-day national conference on Assumed Knowledge in Maths: Its Broad Impact on Tertiary STEM Programs, being hosted at the University of Sydney.

The event acknowledges ongoing concern from teachers and academics about the inadequate mathematical preparation of Australian school students, especially when compared to their international counterparts.

The event will also look at what this means both for their career opportunities and the Australian economy.

Critical in all science study

Maths is critical to STEM training and careers in these areas because of the way it develops our abilities to conceptualise and solve challenging problems.

It is an essential tool in almost every area of science. This is perhaps easy to understand in physics and chemistry which fundamentally rely on maths, and for psychology which is critically dependent on statistics. But mathematics is used extensively in all the sciences.

The genomic revolution, for example, has changed the nature of the biological sciences and resulted in the dramatic growth of the area of bioinformatics.

Vast amounts of data are now available and only mathematically based approaches are able to extract the patterns from this data. These patterns are informing our understanding of evolution and why different populations have different susceptibilities to diseases such as Type 2 Diabetes and Alzheimer’s disease.

In almost every area of science, technology and engineering some computational modelling is now used to test theories and to develop predictions.

Generic shot of a climate change map. Neil Palmer (CIAT)

Climate modelling is being used to develop predictions on how the earth might respond to the vast amounts of extra energy being trapped in our environment.

The more we learn about cancers, the more we realise that computational models are likely to be the best chance we have of understanding how this complexity functions and how to treat it. This understanding will also inform personalised medicine which guides us in how best to treat an individual’s cancer.

Scientists at our university analyse massive data sets to describe international trade relationships and supply chains in previously unattainable detail.

In everyday lives

We rely on mathematics in many other aspects of our everyday lives. Most of us use smartphones to exchange large amounts of data, some of it confidential. This depends on mathematics, and the field of cryptography, which allows accurate and private sharing of information is growing rapidly.

Smartphones are getting smarter thanks to mathematics. Flickr/Samsung Tomorrow

Quantum computing, which we hope will power future developments in these areas, will require even higher levels of mathematics than current systems.

Mathematical skills are needed from the beginning of any degree in the STEM areas and having to catch up makes the task more challenging than it should be.

It takes time to absorb mathematical concepts so for most students it would be preferable to develop that knowledge over years rather than months.

Impact on careers

The growth in interest in STEM courses is not keeping pace with the growth in careers in these areas and this is further undermined by a lack of proficiency in mathematics. Australia will suffer as a consequence if this disparity is not addressed soon.

Students who undertake sufficient mathematics study in their high school years will find it much easier to undertake STEM courses at university. They will then be able to contribute to, and benefit from, the opportunities that the increasing importance of STEM areas represent for the Australian economy.

What is needed are strategies that address the shortfall in the number of high school teachers actually trained to teach mathematics and science. We also need to address the reality of increasing numbers of underprepared students in STEM courses at university.

How universities contribute to the delivery of these strategies will be one of the topics at this national conference.

Adam Spencer, new Mathematics and Science Ambassador. University of Sydney

As a first step we’re announcing today that Adam Spencer has agreed to serve as the University of Sydney’s Mathematics and Science Ambassador.

He’s a well known media personality with an Honours degree in pure mathematics. So it’s hoped he will help us to inspire students to realise the enjoyment and possibilities that mathematics has to offer.

Trevor Hambley is the Dean of Science at the University of Sydney. He receives funding from the ARC.


This article was originally published on The Conversation. Read the original article.